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Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e., $$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$ Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for ...

I posted this question on Math Stack Exchange, but there were no helpful comments or answers https://math.stackexchange.com/q/4874446/1298448 How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...

When I asked for integrals involving the tribonacci constant $T$, user @nospoon gave the nice answer, $$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}} $$ However, the ...

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...